Solving Linear Equations $-7(x-2)+4=6(x+5)$ Algebraically

In the realm of mathematics, linear equations form the bedrock of numerous problem-solving endeavors. Mastering the art of solving these equations is paramount for students and professionals alike. This guide delves into the intricacies of solving the linear equation $-7(x-2)+4=6(x+5)$ algebraically, providing a step-by-step approach to arrive at the solution. We will explore the fundamental principles that underpin the process, ensuring a thorough understanding of the underlying concepts. By adhering to these principles, you can confidently tackle a wide array of linear equations and cultivate your mathematical prowess. This article is about how to solve linear equations, focusing on the algebraic solution of the equation $-7(x-2)+4=6(x+5)$. Understanding the steps involved in solving algebraic equations is crucial for mathematical proficiency. We aim to provide a step-by-step guide that simplifies the process of solving for x in such equations.

Step 1 The Distributive Property Unveiled

The cornerstone of our approach lies in the application of the distributive property. This fundamental principle dictates how we handle expressions enclosed within parentheses. When confronted with an expression like $-7(x-2)$, the distributive property mandates that we multiply the term outside the parentheses (-7) by each term inside the parentheses (x and -2). Similarly, for the expression $6(x+5)$, we multiply 6 by both x and 5. This initial step is crucial as it sets the stage for simplifying the equation and isolating the variable x. Applying the distributive property correctly is the first step in simplifying equations. This involves multiplying the term outside the parentheses by each term inside. For the equation $-7(x-2)+4=6(x+5)$, we first distribute -7 and 6 across their respective parentheses. This foundational step is essential for solving for variables in algebraic expressions and ensures we handle the equation’s components accurately. Remember, the correct application of the distributive property is key to avoiding common errors and achieving a simplified form of the equation.

Let's put this into action:

• $-7(x-2) = -7 * x + (-7) * (-2) = -7x + 14$
• $6(x+5) = 6 * x + 6 * 5 = 6x + 30$

Now, our equation transforms into:

$-7x + 14 + 4 = 6x + 30$

Step 2 The Art of Combining Like Terms

The next critical step involves the judicious combination of like terms. Like terms are those that share the same variable raised to the same power. In our equation, $-7x + 14 + 4 = 6x + 30$, the constants 14 and 4 are like terms, and they can be combined. Similarly, if we had terms like 2x and 3x, they would also be considered like terms. Combining like terms streamlines the equation, making it more manageable and paving the way for isolating the variable. The process of combining like terms is a fundamental step in equation simplification. This involves adding or subtracting terms that have the same variable and exponent. In our equation, $-7x + 14 + 4 = 6x + 30$, the constants 14 and 4 can be combined. Streamlining the equation by combining like terms makes it easier to isolate the variable. Mastering this step is essential for efficient algebraic problem-solving. The ability to recognize and combine like terms is a cornerstone of algebraic manipulation, enabling a clearer path to the solution.

In our equation, we combine 14 and 4:

$14 + 4 = 18$

Our equation now simplifies to:

$-7x + 18 = 6x + 30$

Step 3 Isolating the Variable A Strategic Maneuver

The crux of solving for x lies in isolating it on one side of the equation. To achieve this, we employ inverse operations. Inverse operations are those that undo each other; addition and subtraction are inverse operations, as are multiplication and division. Our aim is to manipulate the equation in such a way that all terms containing x are on one side and all constant terms are on the other. This strategic maneuver brings us closer to unveiling the value of x. Isolating the variable is the key to solving for x in any algebraic equation. This involves using inverse operations to move terms around the equation. The goal is to get all terms containing x on one side and all constants on the other. This strategic manipulation is a crucial step in simplifying the equation and revealing the value of x. Effective variable isolation requires a clear understanding of inverse operations and their application in maintaining the equation’s balance.

Let's move all x terms to one side. We can add 7x to both sides of the equation:

$-7x + 18 + 7x = 6x + 30 + 7x$

This simplifies to:

$18 = 13x + 30$

Now, let's move the constant term to the other side by subtracting 30 from both sides:

$18 - 30 = 13x + 30 - 30$

This simplifies to:

$-12 = 13x$

Step 4 Unveiling the Solution The Final Division

With the variable term isolated, the final step involves dividing both sides of the equation by the coefficient of x. The coefficient of x is the number that multiplies x. In our equation, the coefficient of x is 13. Dividing both sides by 13 will isolate x and reveal its value, thus providing the solution to the equation. This decisive step brings us to the culmination of our algebraic journey. The final step in solving for x involves dividing both sides of the equation by the coefficient of x. This isolates x and reveals its value, providing the solution to the equation. In our case, after isolating the variable term, we divide by the coefficient to find x. This final division is a critical step in algebraic problem-solving, ensuring we arrive at the correct value for x. It represents the culmination of our algebraic manipulations and confirms our solution.

Dividing both sides by 13, we get:

$x = -12/13$

Step 5 The Reduced Fractional Form

The question explicitly asks for the answer in reduced, fractional form. This means that the fraction should be simplified to its lowest terms. To achieve this, we look for the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. In our case, the fraction is -12/13. The GCD of 12 and 13 is 1, which means the fraction is already in its simplest form. Therefore, our final answer remains -12/13. Expressing the solution in reduced fractional form is often a requirement in mathematical problems. This involves simplifying the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). For the fraction -12/13, the GCD is 1, indicating it’s already in simplest form. Understanding how to reduce fractions is crucial for providing complete and accurate solutions. This ensures the answer is presented in its most concise and understandable form, meeting the specific requirements of the problem.

Conclusion: Mastering the Algebraic Solution

In this comprehensive guide, we have meticulously dissected the process of solving the linear equation $-7(x-2)+4=6(x+5)$ algebraically. We embarked on this journey by applying the distributive property, paving the way for simplification. We then combined like terms, streamlining the equation and making it more amenable to manipulation. The pivotal step of isolating the variable followed, where we strategically employed inverse operations to segregate x on one side of the equation. Finally, we divided both sides by the coefficient of x, unveiling the solution in its reduced, fractional form. Mastering these steps is not merely about solving a single equation; it's about cultivating a robust understanding of algebraic principles that will serve as a cornerstone for tackling more complex mathematical challenges. The ability to solve algebraic equations is a critical skill in mathematics. By understanding the step-by-step process, we can confidently approach complex problems. Mastering these steps provides a strong foundation for advanced mathematical concepts. This comprehensive guide aims to equip readers with the tools necessary to tackle a wide range of algebraic challenges. The key to success lies in consistent practice and a solid grasp of fundamental principles.

Therefore,

$x = -12/13$